Transcription:---**Topic: Linear Algebra - Change of Basis**Consider the following two ordered bases of $ \mathbb{R}^2 $:$ B = \{(-1, -1), (3, 2)\} $$ C = \{(1, 1), (2, -1)\} $**Problem:**a) Find the change of basis matrix from basis $ B $ to $ C $.b) Find the change of basis matrix from basis $ C $ to $ B $.---The content addresses the concept of change of basis in the context of vector spaces. The problem involves computing the transition matrices between given sets of basis vectors in $ \mathbb{R}^2 $.

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Description: Transcription:---**Topic: Linear Algebra - Change of Basis**Consider the following two ordered bases of $ \mathbb{R}^2 $:$ B = \{(-1, -1), (3, 2)\} $$ C = \{(1, 1), (2, -1)\} $**Problem:**a) Find the change of basis matrix from basis $ B $ t

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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**Topic: Linear Algebra - Change of Basis**

Consider the following two ordered bases of $ \mathbb{R}^2 $:

$ B = \{(-1, -1), (3, 2)\} $

$ C = \{(1, 1), (2, -1)\} $

**Problem:**

a) Find the change of basis matrix from basis $ B $ to $ C $.

b) Find the change of basis matrix from basis $ C $ to $ B $.

---

The content addresses the concept of change of basis in the context of vector spaces. The problem involves computing the transition matrices between given sets of basis vectors in $ \mathbb{R}^2 $.

Expert Solution

Step 1

Given : $B = \{(- 1, - 1), (3, 2)\}, C = \{(1, 1), (- 2, - 1)\}$

First we write basis elements of $B$ as linear combination of elements of $C$

Let

$(- 1, - 1) = a_{1} (1, 1) + a_{2} (- 2, - 1)$ , where $a_{1}, a_{2} \in ℝ$

Then ,

$(- 1, - 1) = (a_{1} - 2 a_{2}, a_{1} - a_{2}) a_{1} - 2 a_{2} = - 1, a_{1} - a_{2} = - 1$

Solving the above two equations, we get

$a_{1} = - 1, a_{2} = 0$

So,

${[(- 1, - 1)]}_{C} = [\begin{matrix} - 1 \\ 0 \end{matrix}]$

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